Finite Group Databasebeta

← #70128

Subgroups — #70128

Order 384.

230 conjugacy classes of subgroups; only the 65 normal subgroups are shown.

normal subgroups only

This group has 230 conjugacy classes of subgroups — too many to draw. Only its 65 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Conjugacy classes of subgroups

#subgroupG/Hclass sizemultiplicitynormal?properties
1#5816C211yesmaximal, characteristic
2C2×C3×#101C211yesmaximal, characteristic, complemented
3C2×#836C211yesmaximal, characteristic, complemented
4#4478C212yesmaximal, complemented
5#4477C212yesmaximal, complemented
6#2133—31nomaximal, Sylow
7C22×C3×Q8C2211yescharacteristic
8C2×#235C411yescharacteristic
9C2×C3×M16—21no—
10#947C411yescharacteristic
11#947C2211yescharacteristic
12#946—21no—
13#946—21no—
14#944C411yescharacteristic
15#944C2211yescharacteristic
16#929—21no—
17#929—21no—
18#926C411yescharacteristic
19C2×#208—21no—
20C3×#101—21no—
21C3×#101C2212yes—
22#836—21no—
23#836C2212yes—
24#536—31no—
25C2×#101—31no—
26C2×#101D311yescharacteristic, complemented
27#356—32no—
28#355—32no—
29C2×C3×Q8—41no—
30C2×C3×Q8—41no—
31C2×C3×Q8C2×C411yescharacteristic
32C2×C3×Q8D411yescharacteristic
33C2×C3×Q8D411yescharacteristic
34C2×C3×Q8C2311yescharacteristic
35C22×C3×C4C2×C411yescharacteristic
36C22×C3×C4D411yescharacteristic
37C22×C3×C4D411yescharacteristic
38C2×#67—21no—
39#235—41no—
40#235—41no—
41C2×D12C2×C411yescharacteristic
42C2×C4×D3—21no—
43C2×Dic6C2×C411yescharacteristic
44C3×M16—22no—
45C3×M16—41no—
46C2×C3×C8—41no—
47#217—21no—
48(C2×D3)⋊C4—21no—
49(C2×D3)⋊C4—21no—
50(C2×D3)⋊C4—21no—
51(C2×D3)⋊C4—21no—
52C4⋊Dic3—21no—
53Dic3⋊C4—21no—
54Dic3⋊C4—21no—
55Dic3⋊C4—21no—
56Dic3⋊C4—21no—
57C4×Dic3C2×C411yescharacteristic
58C4×Dic3C2×C411yescharacteristic
59#208—22no—
60#208—41no—
61C2×C3⋊C8—41no—
62C2×#41—31no—
63C22×Q8C2×D311yescharacteristic
64C2×M16—21no—
65C2×M16—61no—
66C4⋊Q8—31no—
67C4⋊Q8—31no—
68#124—31no—
69#124—31no—
70#123—61no—
71#123—61no—
72C22⋊Q8—61no—
73C22⋊Q8—61no—
74#117—31no—
75#101—21no—
76#101—32no—
77#101—61no—
78#101C2×D312yes—
79C23×C3C22⋊C411yescharacteristic
80C22×D3—21no—
81C3×Q8—41no—
82C3×Q8—41no—
83C3×Q8—41no—
84C3×Q8—41no—
85C2×C3×C4—21no—
86C2×C3×C4—21no—
87C2×C3×C4—21no—
88C2×C3×C4—21no—
89C2×C3×C4—21no—
90C2×C3×C4—21no—
91C2×C3×C4C22⋊C411yescharacteristic
92C2×C3×C4C22⋊C411yescharacteristic
93C2×C3×C4C22⋊C411yescharacteristic
94C2×C3×C4C22×C411yescharacteristic
95C2×C3×C4C2×D411yescharacteristic
96C2×C3×C4C2×D411yescharacteristic
97#67—41no—
98#67—41no—
99C2×Dic3—21no—
100C2×Dic3—21no—
101C2×Dic3—21no—
102D12—41no—
103C4×D3—42no—
104Dic6—41no—
105C3×C8—42no—
106C3⋊C8—42no—
107#41—121no—
108#41—121no—
109C2×Q8—31no—
110C2×Q8—41no—
111C2×Q8—41no—
112C2×Q8C4×D311yescharacteristic
113C2×Q8D1211yescharacteristic
114C2×Q8#6711yescharacteristic
115C2×Q8C22×D311yescharacteristic
116C2×D4—31no—
117C2×D4—61no—
118C22×C4—61no—
119C22×C4C4×D311yescharacteristic
120C22×C4D1211yescharacteristic
121C22×C4#6711yescharacteristic
122M16—22no—
123M16—41no—
124M16—62no—
125M16—121no—
126C2×C8—41no—
127C2×C8—121no—
128C4⋊C4—61no—
129C4⋊C4—61no—
130C4⋊C4—61no—
131C4⋊C4—61no—
132C4⋊C4—61no—
133C22⋊C4—61no—
134C22⋊C4—61no—
135C22⋊C4—61no—
136C22⋊C4—61no—
137C22⋊C4—61no—
138C42—31no—
139C42—31no—
140C22×C3—41no—
141C22×C3#9911yescharacteristic
142C22×C3#9911yescharacteristic
143C22×C3C2×C22⋊C411yescharacteristic
144C2×D3—42no—
145C2×D3—41no—
146C3×C4—21no—
147C3×C4—21no—
148C3×C4—21no—
149C3×C4—21no—
150C3×C4—21no—
151C3×C4—21no—
152Dic3—41no—
153Dic3—41no—
154Dic3—41no—
155C23—61no—
156C23(C2×D3)⋊C411yescharacteristic
157Q8—41no—
158Q8—41no—
159Q8—41no—
160Q8—41no—
161Q8—121no—
162D4—121no—
163D4—121no—
164D4—121no—
165C2×C4—21no—
166C2×C4—21no—
167C2×C4—21no—
168C2×C4—21no—
169C2×C4—21no—
170C2×C4—21no—
171C2×C4—61no—
172C2×C4—61no—
173C2×C4—61no—
174C2×C4—122no—
175C2×C4(C2×D3)⋊C411yescharacteristic
176C2×C4(C2×D3)⋊C411yescharacteristic
177C2×C4(C2×D3)⋊C411yescharacteristic
178C2×C4C2×C4×D311yescharacteristic
179C2×C4C2×D1211yescharacteristic
180C2×C4C2×#6711yescharacteristic
181C8—42no—
182C8—122no—
183C2×C3—21no—
184C2×C3—21no—
185C2×C3—21no—
186C2×C3C2×#9911yescharacteristic
187D3—81no—
188C22—41no—
189C22—122no—
190C22—121no—
191C22#80611yescharacteristic
192C22#80611yescharacteristic
193C22C2×(C2×D3)⋊C411yescharacteristic
194C4—21no—
195C4—21no—
196C4—21no—
197C4—21no—
198C4—21no—
199C4—21no—
200C4—121no—
201C4—121no—
202C4—121no—
203C3#213311yescharacteristic, complemented, Sylow
204C2—21no—
205C2—21no—
206C2—21no—
207C2—241no—
208C2C2×#80611yescharacteristic
subgroupG/H
centerC2C2×#806
commutatorC2×C3×C4C22×C4
FrattiniC2×Q8C22×D3
FittingC2×C3×#101C2
socleC2×C3C2×#99
prime radicalC2×#235C4
cosocleC2×C3×Q8C23
hypercenterC2×#101D3
hypocenterC3#2133
LCSG ▷ C2×C3×C4 ▷ C22×C3 ▷ C2×C3 ▷ C3
UCSC1 ◁ C2 ◁ C23 ◁ C22×Q8 ◁ C2×#101

Complements

Splittings G = N ⋊ H with N normal; H ≅ G/N.

NH ≅ G/Ntype
2. C2×C3×#101C2semidirect
3. C2×#836C2semidirect
4. #4478C2semidirect
5. #4477C2semidirect
26. C2×#101D3semidirect
203. C3#2133semidirect

Maximal subgroups

By isomorphism type — maximal subgroups of the same type appear once, whether or not they are normal.

subgroupindex
#58162
C2×C3×#1012
C2×#8362
#44782
#44772
#21333